Analytical solutions to the isothermal Euler equations with time-dependent damping

Abstract

In this paper, we construct a class of analytical solutions to the one dimensional compressible isothermal Euler equations with time-dependent damping. By introducing a special density function ρ(x, t) = e c(t)x+d(t), we obtain a family of analytical solutions. A sufficient condition for the solution to blow up in finite time is given. On the basis of the conclusion, the analytical solutions to the initial boundary value problem of the pressureless Euler equations are obtained.